math.sqrt on complex, imaginary part, im greater than 0 branch

Percentage Accurate: 41.0% → 89.9%

Time: 7.9s

Alternatives: 8

Speedup: 2.1×

Specification

\[im > 0\]

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) - re)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 41.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) - re)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 89.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \leq 0:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* re re) (* im im))) re) 0.0)
   (/ (* im 0.5) (sqrt re))
   (sqrt (* 0.5 (- (hypot re im) re)))))

C

double code(double re, double im) {
	double tmp;
	if ((sqrt(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = (im * 0.5) / sqrt(re);
	} else {
		tmp = sqrt((0.5 * (hypot(re, im) - re)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.sqrt(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = (im * 0.5) / Math.sqrt(re);
	} else {
		tmp = Math.sqrt((0.5 * (Math.hypot(re, im) - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.sqrt(((re * re) + (im * im))) - re) <= 0.0:
		tmp = (im * 0.5) / math.sqrt(re)
	else:
		tmp = math.sqrt((0.5 * (math.hypot(re, im) - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re) <= 0.0)
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	else
		tmp = sqrt(Float64(0.5 * Float64(hypot(re, im) - re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((sqrt(((re * re) + (im * im))) - re) <= 0.0)
		tmp = (im * 0.5) / sqrt(re);
	else
		tmp = sqrt((0.5 * (hypot(re, im) - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision], 0.0], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 8.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around inf 88.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]
   4. Step-by-step derivation

      1. associate-*l* 88.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

      2. *-commutative 88.1%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5}\right)} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

   5. Simplified 88.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 88.2%

         \[\leadsto \color{blue}{\left(0.5 \cdot im\right) \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)} \]

      2. sqrt-unprod 89.1%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \left(\color{blue}{\sqrt{2 \cdot 0.5}} \cdot \sqrt{\frac{1}{re}}\right) \]

      3. metadata-eval 89.1%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \left(\sqrt{\color{blue}{1}} \cdot \sqrt{\frac{1}{re}}\right) \]

      4. metadata-eval 89.1%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \left(\color{blue}{1} \cdot \sqrt{\frac{1}{re}}\right) \]

      5. *-un-lft-identity 89.1%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \color{blue}{\sqrt{\frac{1}{re}}} \]

      6. sqrt-div 89.1%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}} \]

      7. metadata-eval 89.1%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \frac{\color{blue}{1}}{\sqrt{re}} \]

      8. un-div-inv 89.2%

         \[\leadsto \color{blue}{\frac{0.5 \cdot im}{\sqrt{re}}} \]

      9. *-commutative 89.2%

         \[\leadsto \frac{\color{blue}{im \cdot 0.5}}{\sqrt{re}} \]

   7. Applied egg-rr 89.2%

      \[\leadsto \color{blue}{\frac{im \cdot 0.5}{\sqrt{re}}} \]

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)

   1. Initial program 47.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 46.8%

         \[\leadsto \color{blue}{\sqrt{0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \cdot \sqrt{0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}} \]

      2. sqrt-unprod 47.1%

         \[\leadsto \color{blue}{\sqrt{\left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}} \]

      3. *-commutative 47.1%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right)} \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)} \]

      4. *-commutative 47.1%

         \[\leadsto \sqrt{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right)}} \]

      5. swap-sqr 47.1%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \left(0.5 \cdot 0.5\right)}} \]

      6. add-sqr-sqrt 47.1%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

      7. *-commutative 47.1%

         \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{re \cdot re + im \cdot im} - re\right) \cdot 2\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

      8. hypot-define 90.9%

         \[\leadsto \sqrt{\left(\left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right) \cdot 2\right) \cdot \left(0.5 \cdot 0.5\right)} \]

      9. metadata-eval 90.9%

         \[\leadsto \sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot \color{blue}{0.25}} \]

   4. Applied egg-rr 90.9%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot 0.25}} \]
   5. Step-by-step derivation

      1. associate-*l* 91.4%

         \[\leadsto \sqrt{\color{blue}{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(2 \cdot 0.25\right)}} \]

      2. metadata-eval 91.4%

         \[\leadsto \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \color{blue}{0.5}} \]

   6. Simplified 91.4%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \leq 0:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.52 \cdot 10^{-33}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.26 \cdot 10^{-61}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.52e-33)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 1.26e-61)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* im (* 0.5 (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.52e-33) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 1.26e-61) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = im * (0.5 * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.52d-33)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 1.26d-61) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = im * (0.5d0 * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.52e-33) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 1.26e-61) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = im * (0.5 * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.52e-33:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 1.26e-61:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = im * (0.5 * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.52e-33)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 1.26e-61)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(im * Float64(0.5 * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.52e-33)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 1.26e-61)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = im * (0.5 * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.52e-33], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.26e-61], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.52e-33

   1. Initial program 42.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. sub-neg 42.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} + \left(-re\right)\right)}} \]

      2. sqr-neg 42.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + \color{blue}{\left(-im\right) \cdot \left(-im\right)}} + \left(-re\right)\right)} \]

      3. sub-neg 42.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + \left(-im\right) \cdot \left(-im\right)} - re\right)}} \]

      4. sqr-neg 42.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}} - re\right)} \]

      5. hypot-define 98.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in re around -inf 78.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   6. Step-by-step derivation

      1. *-commutative 78.2%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   7. Simplified 78.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -1.52e-33 < re < 1.2599999999999999e-61

   1. Initial program 58.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 80.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im} - re\right)} \]

   if 1.2599999999999999e-61 < re

   1. Initial program 16.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 16.3%

         \[\leadsto \color{blue}{\sqrt{0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \cdot \sqrt{0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}} \]

      2. sqrt-unprod 16.4%

         \[\leadsto \color{blue}{\sqrt{\left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}} \]

      3. *-commutative 16.4%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right)} \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)} \]

      4. *-commutative 16.4%

         \[\leadsto \sqrt{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right)}} \]

      5. swap-sqr 16.4%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \left(0.5 \cdot 0.5\right)}} \]

      6. add-sqr-sqrt 16.4%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

      7. *-commutative 16.4%

         \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{re \cdot re + im \cdot im} - re\right) \cdot 2\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

      8. hypot-define 48.0%

         \[\leadsto \sqrt{\left(\left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right) \cdot 2\right) \cdot \left(0.5 \cdot 0.5\right)} \]

      9. metadata-eval 48.0%

         \[\leadsto \sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot \color{blue}{0.25}} \]

   4. Applied egg-rr 48.0%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot 0.25}} \]
   5. Step-by-step derivation

      1. associate-*l* 48.0%

         \[\leadsto \sqrt{\color{blue}{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(2 \cdot 0.25\right)}} \]

      2. metadata-eval 48.0%

         \[\leadsto \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \color{blue}{0.5}} \]

   6. Simplified 48.0%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 0.5}} \]

   7. Taylor expanded in re around inf 65.4%

      \[\leadsto \color{blue}{\left(im \cdot {\left(\sqrt{0.5}\right)}^{2}\right) \cdot \sqrt{\frac{1}{re}}} \]
   8. Step-by-step derivation

      1. associate-*l* 65.3%

         \[\leadsto \color{blue}{im \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot \sqrt{\frac{1}{re}}\right)} \]

      2. unpow2 65.3%

         \[\leadsto im \cdot \left(\color{blue}{\left(\sqrt{0.5} \cdot \sqrt{0.5}\right)} \cdot \sqrt{\frac{1}{re}}\right) \]

      3. rem-square-sqrt 66.1%

         \[\leadsto im \cdot \left(\color{blue}{0.5} \cdot \sqrt{\frac{1}{re}}\right) \]

      4. rem-exp-log 62.8%

         \[\leadsto im \cdot \left(0.5 \cdot \sqrt{\frac{1}{\color{blue}{e^{\log re}}}}\right) \]

      5. exp-neg 62.8%

         \[\leadsto im \cdot \left(0.5 \cdot \sqrt{\color{blue}{e^{-\log re}}}\right) \]

      6. unpow1/2 62.8%

         \[\leadsto im \cdot \left(0.5 \cdot \color{blue}{{\left(e^{-\log re}\right)}^{0.5}}\right) \]

      7. exp-prod 62.8%

         \[\leadsto im \cdot \left(0.5 \cdot \color{blue}{e^{\left(-\log re\right) \cdot 0.5}}\right) \]

      8. distribute-lft-neg-out 62.8%

         \[\leadsto im \cdot \left(0.5 \cdot e^{\color{blue}{-\log re \cdot 0.5}}\right) \]

      9. distribute-rgt-neg-in 62.8%

         \[\leadsto im \cdot \left(0.5 \cdot e^{\color{blue}{\log re \cdot \left(-0.5\right)}}\right) \]

      10. metadata-eval 62.8%

          \[\leadsto im \cdot \left(0.5 \cdot e^{\log re \cdot \color{blue}{-0.5}}\right) \]

      11. exp-to-pow 66.2%

          \[\leadsto im \cdot \left(0.5 \cdot \color{blue}{{re}^{-0.5}}\right) \]

   9. Simplified 66.2%

      \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot {re}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 75.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.15 \cdot 10^{-33}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.15e-33)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 6.2e-67) (sqrt (* 0.5 (- im re))) (* im (* 0.5 (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.15e-33) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 6.2e-67) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = im * (0.5 * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.15d-33)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 6.2d-67) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = im * (0.5d0 * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.15e-33) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 6.2e-67) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = im * (0.5 * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.15e-33:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 6.2e-67:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = im * (0.5 * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.15e-33)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 6.2e-67)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(im * Float64(0.5 * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.15e-33)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 6.2e-67)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = im * (0.5 * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.15e-33], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.2e-67], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(im * N[(0.5 * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.14999999999999993e-33

   1. Initial program 42.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. sub-neg 42.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} + \left(-re\right)\right)}} \]

      2. sqr-neg 42.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + \color{blue}{\left(-im\right) \cdot \left(-im\right)}} + \left(-re\right)\right)} \]

      3. sub-neg 42.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + \left(-im\right) \cdot \left(-im\right)} - re\right)}} \]

      4. sqr-neg 42.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}} - re\right)} \]

      5. hypot-define 98.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in re around -inf 78.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   6. Step-by-step derivation

      1. *-commutative 78.2%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   7. Simplified 78.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -1.14999999999999993e-33 < re < 6.2000000000000005e-67

   1. Initial program 58.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 58.0%

         \[\leadsto \color{blue}{\sqrt{0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \cdot \sqrt{0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}} \]

      2. sqrt-unprod 58.4%

         \[\leadsto \color{blue}{\sqrt{\left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}} \]

      3. *-commutative 58.4%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right)} \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)} \]

      4. *-commutative 58.4%

         \[\leadsto \sqrt{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right)}} \]

      5. swap-sqr 58.4%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \left(0.5 \cdot 0.5\right)}} \]

      6. add-sqr-sqrt 58.4%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

      7. *-commutative 58.4%

         \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{re \cdot re + im \cdot im} - re\right) \cdot 2\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

      8. hypot-define 89.8%

         \[\leadsto \sqrt{\left(\left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right) \cdot 2\right) \cdot \left(0.5 \cdot 0.5\right)} \]

      9. metadata-eval 89.8%

         \[\leadsto \sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot \color{blue}{0.25}} \]

   4. Applied egg-rr 89.8%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot 0.25}} \]
   5. Step-by-step derivation

      1. associate-*l* 89.8%

         \[\leadsto \sqrt{\color{blue}{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(2 \cdot 0.25\right)}} \]

      2. metadata-eval 89.8%

         \[\leadsto \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \color{blue}{0.5}} \]

   6. Simplified 89.8%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 0.5}} \]

   7. Taylor expanded in re around 0 80.0%

      \[\leadsto \sqrt{\color{blue}{\left(im + -1 \cdot re\right)} \cdot 0.5} \]
   8. Step-by-step derivation

      1. neg-mul-1 80.0%

         \[\leadsto \sqrt{\left(im + \color{blue}{\left(-re\right)}\right) \cdot 0.5} \]

      2. unsub-neg 80.0%

         \[\leadsto \sqrt{\color{blue}{\left(im - re\right)} \cdot 0.5} \]

   9. Simplified 80.0%

      \[\leadsto \sqrt{\color{blue}{\left(im - re\right)} \cdot 0.5} \]

   if 6.2000000000000005e-67 < re

   1. Initial program 16.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 16.3%

         \[\leadsto \color{blue}{\sqrt{0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \cdot \sqrt{0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}} \]

      2. sqrt-unprod 16.4%

         \[\leadsto \color{blue}{\sqrt{\left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}} \]

      3. *-commutative 16.4%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right)} \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)} \]

      4. *-commutative 16.4%

         \[\leadsto \sqrt{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right)}} \]

      5. swap-sqr 16.4%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \left(0.5 \cdot 0.5\right)}} \]

      6. add-sqr-sqrt 16.4%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

      7. *-commutative 16.4%

         \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{re \cdot re + im \cdot im} - re\right) \cdot 2\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

      8. hypot-define 48.0%

         \[\leadsto \sqrt{\left(\left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right) \cdot 2\right) \cdot \left(0.5 \cdot 0.5\right)} \]

      9. metadata-eval 48.0%

         \[\leadsto \sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot \color{blue}{0.25}} \]

   4. Applied egg-rr 48.0%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot 0.25}} \]
   5. Step-by-step derivation

      1. associate-*l* 48.0%

         \[\leadsto \sqrt{\color{blue}{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(2 \cdot 0.25\right)}} \]

      2. metadata-eval 48.0%

         \[\leadsto \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \color{blue}{0.5}} \]

   6. Simplified 48.0%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 0.5}} \]

   7. Taylor expanded in re around inf 65.4%

      \[\leadsto \color{blue}{\left(im \cdot {\left(\sqrt{0.5}\right)}^{2}\right) \cdot \sqrt{\frac{1}{re}}} \]
   8. Step-by-step derivation

      1. associate-*l* 65.3%

         \[\leadsto \color{blue}{im \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot \sqrt{\frac{1}{re}}\right)} \]

      2. unpow2 65.3%

         \[\leadsto im \cdot \left(\color{blue}{\left(\sqrt{0.5} \cdot \sqrt{0.5}\right)} \cdot \sqrt{\frac{1}{re}}\right) \]

      3. rem-square-sqrt 66.1%

         \[\leadsto im \cdot \left(\color{blue}{0.5} \cdot \sqrt{\frac{1}{re}}\right) \]

      4. rem-exp-log 62.8%

         \[\leadsto im \cdot \left(0.5 \cdot \sqrt{\frac{1}{\color{blue}{e^{\log re}}}}\right) \]

      5. exp-neg 62.8%

         \[\leadsto im \cdot \left(0.5 \cdot \sqrt{\color{blue}{e^{-\log re}}}\right) \]

      6. unpow1/2 62.8%

         \[\leadsto im \cdot \left(0.5 \cdot \color{blue}{{\left(e^{-\log re}\right)}^{0.5}}\right) \]

      7. exp-prod 62.8%

         \[\leadsto im \cdot \left(0.5 \cdot \color{blue}{e^{\left(-\log re\right) \cdot 0.5}}\right) \]

      8. distribute-lft-neg-out 62.8%

         \[\leadsto im \cdot \left(0.5 \cdot e^{\color{blue}{-\log re \cdot 0.5}}\right) \]

      9. distribute-rgt-neg-in 62.8%

         \[\leadsto im \cdot \left(0.5 \cdot e^{\color{blue}{\log re \cdot \left(-0.5\right)}}\right) \]

      10. metadata-eval 62.8%

          \[\leadsto im \cdot \left(0.5 \cdot e^{\log re \cdot \color{blue}{-0.5}}\right) \]

      11. exp-to-pow 66.2%

          \[\leadsto im \cdot \left(0.5 \cdot \color{blue}{{re}^{-0.5}}\right) \]

   9. Simplified 66.2%

      \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot {re}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.15 \cdot 10^{-33}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.8 \cdot 10^{-33}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5.8e-33)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 3e-65) (sqrt (* 0.5 (- im re))) (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5.8e-33) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 3e-65) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = (im * 0.5) / sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-5.8d-33)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 3d-65) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = (im * 0.5d0) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5.8e-33) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 3e-65) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5.8e-33:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 3e-65:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5.8e-33)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 3e-65)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5.8e-33)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 3e-65)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5.8e-33], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3e-65], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -5.80000000000000005e-33

   1. Initial program 42.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. sub-neg 42.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} + \left(-re\right)\right)}} \]

      2. sqr-neg 42.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + \color{blue}{\left(-im\right) \cdot \left(-im\right)}} + \left(-re\right)\right)} \]

      3. sub-neg 42.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + \left(-im\right) \cdot \left(-im\right)} - re\right)}} \]

      4. sqr-neg 42.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}} - re\right)} \]

      5. hypot-define 98.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in re around -inf 78.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   6. Step-by-step derivation

      1. *-commutative 78.2%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   7. Simplified 78.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -5.80000000000000005e-33 < re < 2.99999999999999998e-65

   1. Initial program 58.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 58.0%

         \[\leadsto \color{blue}{\sqrt{0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \cdot \sqrt{0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}} \]

      2. sqrt-unprod 58.4%

         \[\leadsto \color{blue}{\sqrt{\left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}} \]

      3. *-commutative 58.4%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right)} \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)} \]

      4. *-commutative 58.4%

         \[\leadsto \sqrt{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right)}} \]

      5. swap-sqr 58.4%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \left(0.5 \cdot 0.5\right)}} \]

      6. add-sqr-sqrt 58.4%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

      7. *-commutative 58.4%

         \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{re \cdot re + im \cdot im} - re\right) \cdot 2\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

      8. hypot-define 89.8%

         \[\leadsto \sqrt{\left(\left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right) \cdot 2\right) \cdot \left(0.5 \cdot 0.5\right)} \]

      9. metadata-eval 89.8%

         \[\leadsto \sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot \color{blue}{0.25}} \]

   4. Applied egg-rr 89.8%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot 0.25}} \]
   5. Step-by-step derivation

      1. associate-*l* 89.8%

         \[\leadsto \sqrt{\color{blue}{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(2 \cdot 0.25\right)}} \]

      2. metadata-eval 89.8%

         \[\leadsto \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \color{blue}{0.5}} \]

   6. Simplified 89.8%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 0.5}} \]

   7. Taylor expanded in re around 0 80.0%

      \[\leadsto \sqrt{\color{blue}{\left(im + -1 \cdot re\right)} \cdot 0.5} \]
   8. Step-by-step derivation

      1. neg-mul-1 80.0%

         \[\leadsto \sqrt{\left(im + \color{blue}{\left(-re\right)}\right) \cdot 0.5} \]

      2. unsub-neg 80.0%

         \[\leadsto \sqrt{\color{blue}{\left(im - re\right)} \cdot 0.5} \]

   9. Simplified 80.0%

      \[\leadsto \sqrt{\color{blue}{\left(im - re\right)} \cdot 0.5} \]

   if 2.99999999999999998e-65 < re

   1. Initial program 16.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around inf 65.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]
   4. Step-by-step derivation

      1. associate-*l* 65.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

      2. *-commutative 65.3%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5}\right)} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

   5. Simplified 65.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 65.3%

         \[\leadsto \color{blue}{\left(0.5 \cdot im\right) \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)} \]

      2. sqrt-unprod 66.1%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \left(\color{blue}{\sqrt{2 \cdot 0.5}} \cdot \sqrt{\frac{1}{re}}\right) \]

      3. metadata-eval 66.1%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \left(\sqrt{\color{blue}{1}} \cdot \sqrt{\frac{1}{re}}\right) \]

      4. metadata-eval 66.1%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \left(\color{blue}{1} \cdot \sqrt{\frac{1}{re}}\right) \]

      5. *-un-lft-identity 66.1%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \color{blue}{\sqrt{\frac{1}{re}}} \]

      6. sqrt-div 66.0%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}} \]

      7. metadata-eval 66.0%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \frac{\color{blue}{1}}{\sqrt{re}} \]

      8. un-div-inv 66.1%

         \[\leadsto \color{blue}{\frac{0.5 \cdot im}{\sqrt{re}}} \]

      9. *-commutative 66.1%

         \[\leadsto \frac{\color{blue}{im \cdot 0.5}}{\sqrt{re}} \]

   7. Applied egg-rr 66.1%

      \[\leadsto \color{blue}{\frac{im \cdot 0.5}{\sqrt{re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.8 \cdot 10^{-33}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{-32}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 9.2 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.4e-32)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 9.2e-65) (sqrt (* 0.5 (- im re))) (* im (sqrt (/ 0.25 re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.4e-32) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 9.2e-65) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = im * sqrt((0.25 / re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.4d-32)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 9.2d-65) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = im * sqrt((0.25d0 / re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.4e-32) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 9.2e-65) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = im * Math.sqrt((0.25 / re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.4e-32:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 9.2e-65:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = im * math.sqrt((0.25 / re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.4e-32)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 9.2e-65)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(im * sqrt(Float64(0.25 / re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.4e-32)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 9.2e-65)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = im * sqrt((0.25 / re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.4e-32], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9.2e-65], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(im * N[Sqrt[N[(0.25 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.39999999999999978e-32

   1. Initial program 42.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. sub-neg 42.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} + \left(-re\right)\right)}} \]

      2. sqr-neg 42.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + \color{blue}{\left(-im\right) \cdot \left(-im\right)}} + \left(-re\right)\right)} \]

      3. sub-neg 42.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + \left(-im\right) \cdot \left(-im\right)} - re\right)}} \]

      4. sqr-neg 42.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}} - re\right)} \]

      5. hypot-define 98.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in re around -inf 78.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   6. Step-by-step derivation

      1. *-commutative 78.2%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   7. Simplified 78.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -3.39999999999999978e-32 < re < 9.1999999999999999e-65

   1. Initial program 58.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 58.0%

         \[\leadsto \color{blue}{\sqrt{0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \cdot \sqrt{0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}} \]

      2. sqrt-unprod 58.4%

         \[\leadsto \color{blue}{\sqrt{\left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}} \]

      3. *-commutative 58.4%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right)} \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)} \]

      4. *-commutative 58.4%

         \[\leadsto \sqrt{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right)}} \]

      5. swap-sqr 58.4%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \left(0.5 \cdot 0.5\right)}} \]

      6. add-sqr-sqrt 58.4%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

      7. *-commutative 58.4%

         \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{re \cdot re + im \cdot im} - re\right) \cdot 2\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

      8. hypot-define 89.8%

         \[\leadsto \sqrt{\left(\left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right) \cdot 2\right) \cdot \left(0.5 \cdot 0.5\right)} \]

      9. metadata-eval 89.8%

         \[\leadsto \sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot \color{blue}{0.25}} \]

   4. Applied egg-rr 89.8%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot 0.25}} \]
   5. Step-by-step derivation

      1. associate-*l* 89.8%

         \[\leadsto \sqrt{\color{blue}{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(2 \cdot 0.25\right)}} \]

      2. metadata-eval 89.8%

         \[\leadsto \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \color{blue}{0.5}} \]

   6. Simplified 89.8%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 0.5}} \]

   7. Taylor expanded in re around 0 80.0%

      \[\leadsto \sqrt{\color{blue}{\left(im + -1 \cdot re\right)} \cdot 0.5} \]
   8. Step-by-step derivation

      1. neg-mul-1 80.0%

         \[\leadsto \sqrt{\left(im + \color{blue}{\left(-re\right)}\right) \cdot 0.5} \]

      2. unsub-neg 80.0%

         \[\leadsto \sqrt{\color{blue}{\left(im - re\right)} \cdot 0.5} \]

   9. Simplified 80.0%

      \[\leadsto \sqrt{\color{blue}{\left(im - re\right)} \cdot 0.5} \]

   if 9.1999999999999999e-65 < re

   1. Initial program 16.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around inf 65.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]
   4. Step-by-step derivation

      1. associate-*l* 65.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

      2. *-commutative 65.3%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5}\right)} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

   5. Simplified 65.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 65.3%

         \[\leadsto \color{blue}{\left(0.5 \cdot im\right) \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)} \]

      2. sqrt-unprod 66.1%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \left(\color{blue}{\sqrt{2 \cdot 0.5}} \cdot \sqrt{\frac{1}{re}}\right) \]

      3. metadata-eval 66.1%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \left(\sqrt{\color{blue}{1}} \cdot \sqrt{\frac{1}{re}}\right) \]

      4. metadata-eval 66.1%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \left(\color{blue}{1} \cdot \sqrt{\frac{1}{re}}\right) \]

      5. *-un-lft-identity 66.1%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \color{blue}{\sqrt{\frac{1}{re}}} \]

      6. sqrt-div 66.0%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}} \]

      7. metadata-eval 66.0%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \frac{\color{blue}{1}}{\sqrt{re}} \]

      8. un-div-inv 66.1%

         \[\leadsto \color{blue}{\frac{0.5 \cdot im}{\sqrt{re}}} \]

      9. *-commutative 66.1%

         \[\leadsto \frac{\color{blue}{im \cdot 0.5}}{\sqrt{re}} \]

   7. Applied egg-rr 66.1%

      \[\leadsto \color{blue}{\frac{im \cdot 0.5}{\sqrt{re}}} \]
   8. Step-by-step derivation

      1. associate-/l* 66.0%

         \[\leadsto \color{blue}{im \cdot \frac{0.5}{\sqrt{re}}} \]

   9. Simplified 66.0%

      \[\leadsto \color{blue}{im \cdot \frac{0.5}{\sqrt{re}}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 65.9%

          \[\leadsto im \cdot \color{blue}{\left(\sqrt{\frac{0.5}{\sqrt{re}}} \cdot \sqrt{\frac{0.5}{\sqrt{re}}}\right)} \]

       2. sqrt-unprod 66.0%

          \[\leadsto im \cdot \color{blue}{\sqrt{\frac{0.5}{\sqrt{re}} \cdot \frac{0.5}{\sqrt{re}}}} \]

       3. frac-times 65.9%

          \[\leadsto im \cdot \sqrt{\color{blue}{\frac{0.5 \cdot 0.5}{\sqrt{re} \cdot \sqrt{re}}}} \]

       4. metadata-eval 65.9%

          \[\leadsto im \cdot \sqrt{\frac{\color{blue}{0.25}}{\sqrt{re} \cdot \sqrt{re}}} \]

       5. add-sqr-sqrt 66.1%

          \[\leadsto im \cdot \sqrt{\frac{0.25}{\color{blue}{re}}} \]

   11. Applied egg-rr 66.1%

       \[\leadsto im \cdot \color{blue}{\sqrt{\frac{0.25}{re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{-32}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 9.2 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.35 \cdot 10^{-33}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.35e-33) (* 0.5 (sqrt (* re -4.0))) (sqrt (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.35e-33) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = sqrt((im * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.35d-33)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = sqrt((im * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.35e-33) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = Math.sqrt((im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.35e-33:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = math.sqrt((im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.35e-33)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = sqrt(Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.35e-33)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = sqrt((im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.35e-33], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.3500000000000001e-33

   1. Initial program 42.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. sub-neg 42.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} + \left(-re\right)\right)}} \]

      2. sqr-neg 42.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + \color{blue}{\left(-im\right) \cdot \left(-im\right)}} + \left(-re\right)\right)} \]

      3. sub-neg 42.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + \left(-im\right) \cdot \left(-im\right)} - re\right)}} \]

      4. sqr-neg 42.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}} - re\right)} \]

      5. hypot-define 98.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in re around -inf 78.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   6. Step-by-step derivation

      1. *-commutative 78.2%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   7. Simplified 78.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -2.3500000000000001e-33 < re

   1. Initial program 40.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 40.4%

         \[\leadsto \color{blue}{\sqrt{0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \cdot \sqrt{0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}} \]

      2. sqrt-unprod 40.7%

         \[\leadsto \color{blue}{\sqrt{\left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}} \]

      3. *-commutative 40.7%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right)} \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)} \]

      4. *-commutative 40.7%

         \[\leadsto \sqrt{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right)}} \]

      5. swap-sqr 40.7%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \left(0.5 \cdot 0.5\right)}} \]

      6. add-sqr-sqrt 40.7%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

      7. *-commutative 40.7%

         \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{re \cdot re + im \cdot im} - re\right) \cdot 2\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

      8. hypot-define 72.1%

         \[\leadsto \sqrt{\left(\left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right) \cdot 2\right) \cdot \left(0.5 \cdot 0.5\right)} \]

      9. metadata-eval 72.1%

         \[\leadsto \sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot \color{blue}{0.25}} \]

   4. Applied egg-rr 72.1%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot 0.25}} \]
   5. Step-by-step derivation

      1. associate-*l* 72.1%

         \[\leadsto \sqrt{\color{blue}{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(2 \cdot 0.25\right)}} \]

      2. metadata-eval 72.1%

         \[\leadsto \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \color{blue}{0.5}} \]

   6. Simplified 72.1%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 0.5}} \]

   7. Taylor expanded in re around 0 62.2%

      \[\leadsto \sqrt{\color{blue}{im} \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 54.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 \cdot \left(im - re\right)} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (sqrt (* 0.5 (- im re))))

C

double code(double re, double im) {
	return sqrt((0.5 * (im - re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sqrt((0.5d0 * (im - re)))
end function

Java

public static double code(double re, double im) {
	return Math.sqrt((0.5 * (im - re)));
}

Python

def code(re, im):
	return math.sqrt((0.5 * (im - re)))

Julia

function code(re, im)
	return sqrt(Float64(0.5 * Float64(im - re)))
end

MATLAB

function tmp = code(re, im)
	tmp = sqrt((0.5 * (im - re)));
end

Wolfram

code[re_, im_] := N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 41.1%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 40.8%

      \[\leadsto \color{blue}{\sqrt{0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \cdot \sqrt{0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}} \]

   2. sqrt-unprod 41.1%

      \[\leadsto \color{blue}{\sqrt{\left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}} \]

   3. *-commutative 41.1%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right)} \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)} \]

   4. *-commutative 41.1%

      \[\leadsto \sqrt{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right)}} \]

   5. swap-sqr 41.1%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \left(0.5 \cdot 0.5\right)}} \]

   6. add-sqr-sqrt 41.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

   7. *-commutative 41.1%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{re \cdot re + im \cdot im} - re\right) \cdot 2\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

   8. hypot-define 80.0%

      \[\leadsto \sqrt{\left(\left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right) \cdot 2\right) \cdot \left(0.5 \cdot 0.5\right)} \]

   9. metadata-eval 80.0%

      \[\leadsto \sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot \color{blue}{0.25}} \]

4. Applied egg-rr 80.0%

   \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot 0.25}} \]
5. Step-by-step derivation

   1. associate-*l* 80.4%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(2 \cdot 0.25\right)}} \]

   2. metadata-eval 80.4%

      \[\leadsto \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \color{blue}{0.5}} \]

6. Simplified 80.4%

   \[\leadsto \color{blue}{\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 0.5}} \]

7. Taylor expanded in re around 0 55.0%

   \[\leadsto \sqrt{\color{blue}{\left(im + -1 \cdot re\right)} \cdot 0.5} \]
8. Step-by-step derivation

   1. neg-mul-1 55.0%

      \[\leadsto \sqrt{\left(im + \color{blue}{\left(-re\right)}\right) \cdot 0.5} \]

   2. unsub-neg 55.0%

      \[\leadsto \sqrt{\color{blue}{\left(im - re\right)} \cdot 0.5} \]

9. Simplified 55.0%

   \[\leadsto \sqrt{\color{blue}{\left(im - re\right)} \cdot 0.5} \]

10. Final simplification 55.0%

    \[\leadsto \sqrt{0.5 \cdot \left(im - re\right)} \]
11. Add Preprocessing

Alternative 8: 52.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{im \cdot 0.5} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (sqrt (* im 0.5)))

C

double code(double re, double im) {
	return sqrt((im * 0.5));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sqrt((im * 0.5d0))
end function

Java

public static double code(double re, double im) {
	return Math.sqrt((im * 0.5));
}

Python

def code(re, im):
	return math.sqrt((im * 0.5))

Julia

function code(re, im)
	return sqrt(Float64(im * 0.5))
end

MATLAB

function tmp = code(re, im)
	tmp = sqrt((im * 0.5));
end

Wolfram

code[re_, im_] := N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 41.1%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 40.8%

      \[\leadsto \color{blue}{\sqrt{0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \cdot \sqrt{0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}} \]

   2. sqrt-unprod 41.1%

      \[\leadsto \color{blue}{\sqrt{\left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}} \]

   3. *-commutative 41.1%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right)} \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)} \]

   4. *-commutative 41.1%

      \[\leadsto \sqrt{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\right)}} \]

   5. swap-sqr 41.1%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \left(0.5 \cdot 0.5\right)}} \]

   6. add-sqr-sqrt 41.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

   7. *-commutative 41.1%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{re \cdot re + im \cdot im} - re\right) \cdot 2\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

   8. hypot-define 80.0%

      \[\leadsto \sqrt{\left(\left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right) \cdot 2\right) \cdot \left(0.5 \cdot 0.5\right)} \]

   9. metadata-eval 80.0%

      \[\leadsto \sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot \color{blue}{0.25}} \]

4. Applied egg-rr 80.0%

   \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot 0.25}} \]
5. Step-by-step derivation

   1. associate-*l* 80.4%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(2 \cdot 0.25\right)}} \]

   2. metadata-eval 80.4%

      \[\leadsto \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \color{blue}{0.5}} \]

6. Simplified 80.4%

   \[\leadsto \color{blue}{\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 0.5}} \]

7. Taylor expanded in re around 0 51.9%

   \[\leadsto \sqrt{\color{blue}{im} \cdot 0.5} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))